LMFDB - Embedded newform 6422.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.361.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 7 $ x^3 - x^2 - 6*x + 7
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.28514$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.28514 q^{3} +1.00000 q^{4} -1.22188 q^{5} -2.28514 q^{6} +5.01404 q^{7} +1.00000 q^{8} +2.22188 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.28514 q^{3} +1.00000 q^{4} -1.22188 q^{5} -2.28514 q^{6} +5.01404 q^{7} +1.00000 q^{8} +2.22188 q^{9} -1.22188 q^{10} -2.50702 q^{11} -2.28514 q^{12} +5.01404 q^{14} +2.79216 q^{15} +1.00000 q^{16} +0.285142 q^{17} +2.22188 q^{18} +1.00000 q^{19} -1.22188 q^{20} -11.4578 q^{21} -2.50702 q^{22} -4.95077 q^{23} -2.28514 q^{24} -3.50702 q^{25} +1.77812 q^{27} +5.01404 q^{28} +9.07730 q^{29} +2.79216 q^{30} -9.34841 q^{31} +1.00000 q^{32} +5.72889 q^{33} +0.285142 q^{34} -6.12653 q^{35} +2.22188 q^{36} -10.5703 q^{37} +1.00000 q^{38} -1.22188 q^{40} +0.728895 q^{41} -11.4578 q^{42} +3.01404 q^{43} -2.50702 q^{44} -2.71486 q^{45} -4.95077 q^{46} -7.01404 q^{47} -2.28514 q^{48} +18.1406 q^{49} -3.50702 q^{50} -0.651591 q^{51} +0.207839 q^{53} +1.77812 q^{54} +3.06327 q^{55} +5.01404 q^{56} -2.28514 q^{57} +9.07730 q^{58} -2.44375 q^{59} +2.79216 q^{60} -7.01404 q^{61} -9.34841 q^{62} +11.1406 q^{63} +1.00000 q^{64} +5.72889 q^{66} -8.88750 q^{67} +0.285142 q^{68} +11.3132 q^{69} -6.12653 q^{70} +8.67967 q^{71} +2.22188 q^{72} -11.9015 q^{73} -10.5703 q^{74} +8.01404 q^{75} +1.00000 q^{76} -12.5703 q^{77} -5.14057 q^{79} -1.22188 q^{80} -10.7289 q^{81} +0.728895 q^{82} +12.1867 q^{83} -11.4578 q^{84} -0.348409 q^{85} +3.01404 q^{86} -20.7429 q^{87} -2.50702 q^{88} +2.50702 q^{89} -2.71486 q^{90} -4.95077 q^{92} +21.3624 q^{93} -7.01404 q^{94} -1.22188 q^{95} -2.28514 q^{96} -0.478944 q^{97} +18.1406 q^{98} -5.57028 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + 4 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 - 2 * q^7 + 3 * q^8 + 4 * q^9	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + 4 q^{9} - q^{10} + q^{11} - q^{12} - 2 q^{14} - 6 q^{15} + 3 q^{16} - 5 q^{17} + 4 q^{18} + 3 q^{19} - q^{20} - 12 q^{21} + q^{22} - q^{23} - q^{24} - 2 q^{25} + 8 q^{27} - 2 q^{28} + 7 q^{29} - 6 q^{30} - 19 q^{31} + 3 q^{32} + 6 q^{33} - 5 q^{34} - 12 q^{35} + 4 q^{36} - 20 q^{37} + 3 q^{38} - q^{40} - 9 q^{41} - 12 q^{42} - 8 q^{43} + q^{44} - 14 q^{45} - q^{46} - 4 q^{47} - q^{48} + 31 q^{49} - 2 q^{50} - 11 q^{51} + 15 q^{53} + 8 q^{54} + 6 q^{55} - 2 q^{56} - q^{57} + 7 q^{58} - 2 q^{59} - 6 q^{60} - 4 q^{61} - 19 q^{62} + 10 q^{63} + 3 q^{64} + 6 q^{66} - 16 q^{67} - 5 q^{68} - 6 q^{69} - 12 q^{70} + q^{71} + 4 q^{72} - 8 q^{73} - 20 q^{74} + 7 q^{75} + 3 q^{76} - 26 q^{77} + 8 q^{79} - q^{80} - 21 q^{81} - 9 q^{82} + 3 q^{83} - 12 q^{84} + 8 q^{85} - 8 q^{86} - 34 q^{87} + q^{88} - q^{89} - 14 q^{90} - q^{92} + 38 q^{93} - 4 q^{94} - q^{95} - q^{96} - 27 q^{97} + 31 q^{98} - 5 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 - 2 * q^7 + 3 * q^8 + 4 * q^9 - q^10 + q^11 - q^12 - 2 * q^14 - 6 * q^15 + 3 * q^16 - 5 * q^17 + 4 * q^18 + 3 * q^19 - q^20 - 12 * q^21 + q^22 - q^23 - q^24 - 2 * q^25 + 8 * q^27 - 2 * q^28 + 7 * q^29 - 6 * q^30 - 19 * q^31 + 3 * q^32 + 6 * q^33 - 5 * q^34 - 12 * q^35 + 4 * q^36 - 20 * q^37 + 3 * q^38 - q^40 - 9 * q^41 - 12 * q^42 - 8 * q^43 + q^44 - 14 * q^45 - q^46 - 4 * q^47 - q^48 + 31 * q^49 - 2 * q^50 - 11 * q^51 + 15 * q^53 + 8 * q^54 + 6 * q^55 - 2 * q^56 - q^57 + 7 * q^58 - 2 * q^59 - 6 * q^60 - 4 * q^61 - 19 * q^62 + 10 * q^63 + 3 * q^64 + 6 * q^66 - 16 * q^67 - 5 * q^68 - 6 * q^69 - 12 * q^70 + q^71 + 4 * q^72 - 8 * q^73 - 20 * q^74 + 7 * q^75 + 3 * q^76 - 26 * q^77 + 8 * q^79 - q^80 - 21 * q^81 - 9 * q^82 + 3 * q^83 - 12 * q^84 + 8 * q^85 - 8 * q^86 - 34 * q^87 + q^88 - q^89 - 14 * q^90 - q^92 + 38 * q^93 - 4 * q^94 - q^95 - q^96 - 27 * q^97 + 31 * q^98 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.28514	−1.31933	−0.659664	−	0.751561i	$-0.729301\pi$
$3$	−2.28514	−1.31933	−0.659664	+	0.751561i	$0.729301\pi$
$4$	1.00000	0.500000
$5$	−1.22188	−0.546440	−0.273220	−	0.961952i	$-0.588089\pi$
$5$	−1.22188	−0.546440	−0.273220	+	0.961952i	$0.588089\pi$
$6$	−2.28514	−0.932906
$7$	5.01404	1.89513	0.947564	−	0.319566i	$-0.103537\pi$
$7$	5.01404	1.89513	0.947564	+	0.319566i	$0.103537\pi$
$8$	1.00000	0.353553
$9$	2.22188	0.740625
$10$	−1.22188	−0.386391
$11$	−2.50702	−0.755895	−0.377947	−	0.925827i	$-0.623370\pi$
$11$	−2.50702	−0.755895	−0.377947	+	0.925827i	$0.623370\pi$
$12$	−2.28514	−0.659664
$13$	0	0
$13$	0	0
$14$	5.01404	1.34006
$15$	2.79216	0.720933
$16$	1.00000	0.250000
$17$	0.285142	0.0691572	0.0345786	−	0.999402i	$-0.488991\pi$
$17$	0.285142	0.0691572	0.0345786	+	0.999402i	$0.488991\pi$
$18$	2.22188	0.523701
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.22188	−0.273220
$21$	−11.4578	−2.50029
$22$	−2.50702	−0.534498
$23$	−4.95077	−1.03231	−0.516154	−	0.856496i	$-0.672637\pi$
$23$	−4.95077	−1.03231	−0.516154	+	0.856496i	$0.672637\pi$
$24$	−2.28514	−0.466453
$25$	−3.50702	−0.701404
$26$	0	0
$27$	1.77812	0.342200
$28$	5.01404	0.947564
$29$	9.07730	1.68561	0.842806	−	0.538217i	$-0.180902\pi$
$29$	9.07730	1.68561	0.842806	+	0.538217i	$0.180902\pi$
$30$	2.79216	0.509777
$31$	−9.34841	−1.67902	−0.839512	−	0.543341i	$-0.817159\pi$
$31$	−9.34841	−1.67902	−0.839512	+	0.543341i	$0.817159\pi$
$32$	1.00000	0.176777
$33$	5.72889	0.997273
$34$	0.285142	0.0489015
$35$	−6.12653	−1.03557
$36$	2.22188	0.370313
$37$	−10.5703	−1.73774	−0.868872	−	0.495037i	$-0.835155\pi$
$37$	−10.5703	−1.73774	−0.868872	+	0.495037i	$0.835155\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−1.22188	−0.193196
$41$	0.728895	0.113834	0.0569171	−	0.998379i	$-0.481873\pi$
$41$	0.728895	0.113834	0.0569171	+	0.998379i	$0.481873\pi$
$42$	−11.4578	−1.76798
$43$	3.01404	0.459636	0.229818	−	0.973234i	$-0.426187\pi$
$43$	3.01404	0.459636	0.229818	+	0.973234i	$0.426187\pi$
$44$	−2.50702	−0.377947
$45$	−2.71486	−0.404707
$46$	−4.95077	−0.729951
$47$	−7.01404	−1.02310	−0.511551	−	0.859253i	$-0.670929\pi$
$47$	−7.01404	−1.02310	−0.511551	+	0.859253i	$0.670929\pi$
$48$	−2.28514	−0.329832
$49$	18.1406	2.59151
$50$	−3.50702	−0.495967
$51$	−0.651591	−0.0912410
$52$	0	0
$53$	0.207839	0.0285489	0.0142744	−	0.999898i	$-0.495456\pi$
$53$	0.207839	0.0285489	0.0142744	+	0.999898i	$0.495456\pi$
$54$	1.77812	0.241972
$55$	3.06327	0.413051
$56$	5.01404	0.670029
$57$	−2.28514	−0.302675
$58$	9.07730	1.19191
$59$	−2.44375	−0.318149	−0.159075	−	0.987267i	$-0.550851\pi$
$59$	−2.44375	−0.318149	−0.159075	+	0.987267i	$0.550851\pi$
$60$	2.79216	0.360466
$61$	−7.01404	−0.898055	−0.449028	−	0.893518i	$-0.648230\pi$
$61$	−7.01404	−0.898055	−0.449028	+	0.893518i	$0.648230\pi$
$62$	−9.34841	−1.18725
$63$	11.1406	1.40358
$64$	1.00000	0.125000
$65$	0	0
$66$	5.72889	0.705178
$67$	−8.88750	−1.08578	−0.542891	−	0.839803i	$-0.682670\pi$
$67$	−8.88750	−1.08578	−0.542891	+	0.839803i	$0.682670\pi$
$68$	0.285142	0.0345786
$69$	11.3132	1.36195
$70$	−6.12653	−0.732261
$71$	8.67967	1.03009	0.515043	−	0.857164i	$-0.327776\pi$
$71$	8.67967	1.03009	0.515043	+	0.857164i	$0.327776\pi$
$72$	2.22188	0.261851
$73$	−11.9015	−1.39297	−0.696485	−	0.717571i	$-0.745254\pi$
$73$	−11.9015	−1.39297	−0.696485	+	0.717571i	$0.745254\pi$
$74$	−10.5703	−1.22877
$75$	8.01404	0.925381
$76$	1.00000	0.114708
$77$	−12.5703	−1.43252
$78$	0	0
$79$	−5.14057	−0.578359	−0.289180	−	0.957275i	$-0.593382\pi$
$79$	−5.14057	−0.578359	−0.289180	+	0.957275i	$0.593382\pi$
$80$	−1.22188	−0.136610
$81$	−10.7289	−1.19210
$82$	0.728895	0.0804930
$83$	12.1867	1.33766	0.668831	−	0.743414i	$-0.266795\pi$
$83$	12.1867	1.33766	0.668831	+	0.743414i	$0.266795\pi$
$84$	−11.4578	−1.25015
$85$	−0.348409	−0.0377902
$86$	3.01404	0.325012
$87$	−20.7429	−2.22388
$88$	−2.50702	−0.267249
$89$	2.50702	0.265743	0.132872	−	0.991133i	$-0.457580\pi$
$89$	2.50702	0.265743	0.132872	+	0.991133i	$0.457580\pi$
$90$	−2.71486	−0.286171
$91$	0	0
$92$	−4.95077	−0.516154
$93$	21.3624	2.21518
$94$	−7.01404	−0.723443
$95$	−1.22188	−0.125362
$96$	−2.28514	−0.233226
$97$	−0.478944	−0.0486294	−0.0243147	−	0.999704i	$-0.507740\pi$
$97$	−0.478944	−0.0486294	−0.0243147	+	0.999704i	$0.507740\pi$
$98$	18.1406	1.83247
$99$	−5.57028	−0.559835
$100$	−3.50702	−0.350702
$101$	−0.317220	−0.0315645	−0.0157823	−	0.999875i	$-0.505024\pi$
$101$	−0.317220	−0.0315645	−0.0157823	+	0.999875i	$0.505024\pi$
$102$	−0.651591	−0.0645171
$103$	5.90154	0.581496	0.290748	−	0.956800i	$-0.406096\pi$
$103$	5.90154	0.581496	0.290748	+	0.956800i	$0.406096\pi$
$104$	0	0
$105$	14.0000	1.36626
$106$	0.207839	0.0201871
$107$	−10.2078	−0.986829	−0.493415	−	0.869794i	$-0.664251\pi$
$107$	−10.2078	−0.986829	−0.493415	+	0.869794i	$0.664251\pi$
$108$	1.77812	0.171100
$109$	2.57028	0.246189	0.123094	−	0.992395i	$-0.460718\pi$
$109$	2.57028	0.246189	0.123094	+	0.992395i	$0.460718\pi$
$110$	3.06327	0.292071
$111$	24.1546	2.29265
$112$	5.01404	0.473782
$113$	2.44375	0.229889	0.114944	−	0.993372i	$-0.463331\pi$
$113$	2.44375	0.229889	0.114944	+	0.993372i	$0.463331\pi$
$114$	−2.28514	−0.214023
$115$	6.04923	0.564094
$116$	9.07730	0.842806
$117$	0	0
$118$	−2.44375	−0.224966
$119$	1.42972	0.131062
$120$	2.79216	0.254888
$121$	−4.71486	−0.428623
$122$	−7.01404	−0.635021
$123$	−1.66563	−0.150185
$124$	−9.34841	−0.839512
$125$	10.3945	0.929714
$126$	11.1406	0.992481
$127$	4.15461	0.368662	0.184331	−	0.982864i	$-0.440988\pi$
$127$	4.15461	0.368662	0.184331	+	0.982864i	$0.440988\pi$
$128$	1.00000	0.0883883
$129$	−6.88750	−0.606411
$130$	0	0
$131$	18.1546	1.58618	0.793088	−	0.609107i	$-0.208472\pi$
$131$	18.1546	1.58618	0.793088	+	0.609107i	$0.208472\pi$
$132$	5.72889	0.498636
$133$	5.01404	0.434772
$134$	−8.88750	−0.767763
$135$	−2.17265	−0.186992
$136$	0.285142	0.0244508
$137$	−23.0421	−1.96862	−0.984310	−	0.176446i	$-0.943540\pi$
$137$	−23.0421	−1.96862	−0.984310	+	0.176446i	$0.943540\pi$
$138$	11.3132	0.963045
$139$	16.3453	1.38639	0.693195	−	0.720750i	$-0.256203\pi$
$139$	16.3453	1.38639	0.693195	+	0.720750i	$0.256203\pi$
$140$	−6.12653	−0.517787
$141$	16.0281	1.34981
$142$	8.67967	0.728381
$143$	0	0
$144$	2.22188	0.185156
$145$	−11.0913	−0.921086
$146$	−11.9015	−0.984979
$147$	−41.4538	−3.41905
$148$	−10.5703	−0.868872
$149$	19.8695	1.62777	0.813885	−	0.581026i	$-0.197349\pi$
$149$	19.8695	1.62777	0.813885	+	0.581026i	$0.197349\pi$
$150$	8.01404	0.654343
$151$	−20.3444	−1.65560	−0.827802	−	0.561020i	$-0.810409\pi$
$151$	−20.3444	−1.65560	−0.827802	+	0.561020i	$0.810409\pi$
$152$	1.00000	0.0811107
$153$	0.633551	0.0512196
$154$	−12.5703	−1.01294
$155$	11.4226	0.917485
$156$	0	0
$157$	8.88750	0.709300	0.354650	−	0.934999i	$-0.384600\pi$
$157$	8.88750	0.709300	0.354650	+	0.934999i	$0.384600\pi$
$158$	−5.14057	−0.408962
$159$	−0.474941	−0.0376653
$160$	−1.22188	−0.0965978
$161$	−24.8234	−1.95635
$162$	−10.7289	−0.842942
$163$	−6.72889	−0.527048	−0.263524	−	0.964653i	$-0.584885\pi$
$163$	−6.72889	−0.527048	−0.263524	+	0.964653i	$0.584885\pi$
$164$	0.728895	0.0569171
$165$	−7.00000	−0.544949
$166$	12.1867	0.945870
$167$	0.728895	0.0564036	0.0282018	−	0.999602i	$-0.491022\pi$
$167$	0.728895	0.0564036	0.0282018	+	0.999602i	$0.491022\pi$
$168$	−11.4578	−0.883988
$169$	0	0
$170$	−0.348409	−0.0267217
$171$	2.22188	0.169911
$172$	3.01404	0.229818
$173$	−15.7429	−1.19691	−0.598456	−	0.801155i	$-0.704219\pi$
$173$	−15.7429	−1.19691	−0.598456	+	0.801155i	$0.704219\pi$
$174$	−20.7429	−1.57252
$175$	−17.5843	−1.32925
$176$	−2.50702	−0.188974
$177$	5.58432	0.419743
$178$	2.50702	0.187909
$179$	−12.4890	−0.933470	−0.466735	−	0.884397i	$-0.654570\pi$
$179$	−12.4890	−0.933470	−0.466735	+	0.884397i	$0.654570\pi$
$180$	−2.71486	−0.202354
$181$	−8.88350	−0.660306	−0.330153	−	0.943928i	$-0.607100\pi$
$181$	−8.88350	−0.660306	−0.330153	+	0.943928i	$0.607100\pi$
$182$	0	0
$183$	16.0281	1.18483
$184$	−4.95077	−0.364976
$185$	12.9156	0.949572
$186$	21.3624	1.56637
$187$	−0.714858	−0.0522756
$188$	−7.01404	−0.511551
$189$	8.91558	0.648513
$190$	−1.22188	−0.0886442
$191$	0.553133	0.0400233	0.0200117	−	0.999800i	$-0.493630\pi$
$191$	0.553133	0.0400233	0.0200117	+	0.999800i	$0.493630\pi$
$192$	−2.28514	−0.164916
$193$	19.3273	1.39121	0.695603	−	0.718426i	$-0.255137\pi$
$193$	19.3273	1.39121	0.695603	+	0.718426i	$0.255137\pi$
$194$	−0.478944	−0.0343862
$195$	0	0
$196$	18.1406	1.29575
$197$	2.88750	0.205726	0.102863	−	0.994696i	$-0.467200\pi$
$197$	2.88750	0.205726	0.102863	+	0.994696i	$0.467200\pi$
$198$	−5.57028	−0.395863
$199$	−14.7258	−1.04388	−0.521941	−	0.852981i	$-0.674792\pi$
$199$	−14.7258	−1.04388	−0.521941	+	0.852981i	$0.674792\pi$
$200$	−3.50702	−0.247984
$201$	20.3092	1.43250
$202$	−0.317220	−0.0223195
$203$	45.5139	3.19445
$204$	−0.651591	−0.0456205
$205$	−0.890619	−0.0622035
$206$	5.90154	0.411180
$207$	−11.0000	−0.764553
$208$	0	0
$209$	−2.50702	−0.173414
$210$	14.0000	0.966092
$211$	6.60548	0.454740	0.227370	−	0.973808i	$-0.426987\pi$
$211$	6.60548	0.454740	0.227370	+	0.973808i	$0.426987\pi$
$212$	0.207839	0.0142744
$213$	−19.8343	−1.35902
$214$	−10.2078	−0.697793
$215$	−3.68278	−0.251164
$216$	1.77812	0.120986
$217$	−46.8733	−3.18196
$218$	2.57028	0.174082
$219$	27.1967	1.83778
$220$	3.06327	0.206525
$221$	0	0
$222$	24.1546	1.62115
$223$	−4.41168	−0.295428	−0.147714	−	0.989030i	$-0.547191\pi$
$223$	−4.41168	−0.295428	−0.147714	+	0.989030i	$0.547191\pi$
$224$	5.01404	0.335014
$225$	−7.79216	−0.519477
$226$	2.44375	0.162556
$227$	−25.8655	−1.71675	−0.858376	−	0.513022i	$-0.828526\pi$
$227$	−25.8655	−1.71675	−0.858376	+	0.513022i	$0.828526\pi$
$228$	−2.28514	−0.151337
$229$	−9.93673	−0.656638	−0.328319	−	0.944567i	$-0.606482\pi$
$229$	−9.93673	−0.656638	−0.328319	+	0.944567i	$0.606482\pi$
$230$	6.04923	0.398874
$231$	28.7249	1.88996
$232$	9.07730	0.595954
$233$	−13.5030	−0.884612	−0.442306	−	0.896864i	$-0.645839\pi$
$233$	−13.5030	−0.884612	−0.442306	+	0.896864i	$0.645839\pi$
$234$	0	0
$235$	8.57028	0.559064
$236$	−2.44375	−0.159075
$237$	11.7469	0.763045
$238$	1.42972	0.0926747
$239$	−20.8875	−1.35110	−0.675550	−	0.737314i	$-0.736094\pi$
$239$	−20.8875	−1.35110	−0.675550	+	0.737314i	$0.736094\pi$
$240$	2.79216	0.180233
$241$	−23.5672	−1.51809	−0.759047	−	0.651035i	$-0.774335\pi$
$241$	−23.5672	−1.51809	−0.759047	+	0.651035i	$0.774335\pi$
$242$	−4.71486	−0.303083
$243$	19.1827	1.23057
$244$	−7.01404	−0.449028
$245$	−22.1655	−1.41610
$246$	−1.66563	−0.106197
$247$	0	0
$248$	−9.34841	−0.593625
$249$	−27.8483	−1.76481
$250$	10.3945	0.657407
$251$	−15.8374	−0.999647	−0.499824	−	0.866127i	$-0.666602\pi$
$251$	−15.8374	−0.999647	−0.499824	+	0.866127i	$0.666602\pi$
$252$	11.1406	0.701790
$253$	12.4117	0.780315
$254$	4.15461	0.260683
$255$	0.796164	0.0498577
$256$	1.00000	0.0625000
$257$	20.0000	1.24757	0.623783	−	0.781598i	$-0.285595\pi$
$257$	20.0000	1.24757	0.623783	+	0.781598i	$0.285595\pi$
$258$	−6.88750	−0.428797
$259$	−52.9998	−3.29325
$260$	0	0
$261$	20.1686	1.24841
$262$	18.1546	1.12160
$263$	−21.1726	−1.30556	−0.652781	−	0.757547i	$-0.726398\pi$
$263$	−21.1726	−1.30556	−0.652781	+	0.757547i	$0.726398\pi$
$264$	5.72889	0.352589
$265$	−0.253953	−0.0156002
$266$	5.01404	0.307430
$267$	−5.72889	−0.350603
$268$	−8.88750	−0.542891
$269$	15.1406	0.923137	0.461568	−	0.887105i	$-0.347287\pi$
$269$	15.1406	0.923137	0.461568	+	0.887105i	$0.347287\pi$
$270$	−2.17265	−0.132223
$271$	−14.1827	−0.861537	−0.430768	−	0.902463i	$-0.641757\pi$
$271$	−14.1827	−0.861537	−0.430768	+	0.902463i	$0.641757\pi$
$272$	0.285142	0.0172893
$273$	0	0
$274$	−23.0421	−1.39202
$275$	8.79216	0.530187
$276$	11.3132	0.680976
$277$	9.01404	0.541601	0.270801	−	0.962635i	$-0.412712\pi$
$277$	9.01404	0.541601	0.270801	+	0.962635i	$0.412712\pi$
$278$	16.3453	0.980326
$279$	−20.7710	−1.24353
$280$	−6.12653	−0.366130
$281$	−27.5211	−1.64177	−0.820884	−	0.571095i	$-0.806519\pi$
$281$	−27.5211	−1.64177	−0.820884	+	0.571095i	$0.806519\pi$
$282$	16.0281	0.954458
$283$	−2.95789	−0.175828	−0.0879141	−	0.996128i	$-0.528020\pi$
$283$	−2.95789	−0.175828	−0.0879141	+	0.996128i	$0.528020\pi$
$284$	8.67967	0.515043
$285$	2.79216	0.165393
$286$	0	0
$287$	3.65471	0.215730
$288$	2.22188	0.130925
$289$	−16.9187	−0.995217
$290$	−11.0913	−0.651306
$291$	1.09446	0.0641581
$292$	−11.9015	−0.696485
$293$	−17.6124	−1.02893	−0.514464	−	0.857512i	$-0.672009\pi$
$293$	−17.6124	−1.02893	−0.514464	+	0.857512i	$0.672009\pi$
$294$	−41.4538	−2.41763
$295$	2.98596	0.173849
$296$	−10.5703	−0.614385
$297$	−4.45779	−0.258667
$298$	19.8695	1.15101
$299$	0	0
$300$	8.01404	0.462691
$301$	15.1125	0.871070
$302$	−20.3444	−1.17069
$303$	0.724892	0.0416440
$304$	1.00000	0.0573539
$305$	8.57028	0.490733
$306$	0.633551	0.0362177
$307$	−5.74693	−0.327995	−0.163997	−	0.986461i	$-0.552439\pi$
$307$	−5.74693	−0.327995	−0.163997	+	0.986461i	$0.552439\pi$
$308$	−12.5703	−0.716258
$309$	−13.4859	−0.767184
$310$	11.4226	0.648760
$311$	−15.7149	−0.891108	−0.445554	−	0.895255i	$-0.646993\pi$
$311$	−15.7149	−0.891108	−0.445554	+	0.895255i	$0.646993\pi$
$312$	0	0
$313$	0.306297	0.0173129	0.00865646	−	0.999963i	$-0.497245\pi$
$313$	0.306297	0.0173129	0.00865646	+	0.999963i	$0.497245\pi$
$314$	8.88750	0.501551
$315$	−13.6124	−0.766972
$316$	−5.14057	−0.289180
$317$	2.25307	0.126545	0.0632724	−	0.997996i	$-0.479846\pi$
$317$	2.25307	0.126545	0.0632724	+	0.997996i	$0.479846\pi$
$318$	−0.474941	−0.0266334
$319$	−22.7570	−1.27415
$320$	−1.22188	−0.0683050
$321$	23.3264	1.30195
$322$	−24.8234	−1.38335
$323$	0.285142	0.0158658
$324$	−10.7289	−0.596050
$325$	0	0
$326$	−6.72889	−0.372679
$327$	−5.87347	−0.324803
$328$	0.728895	0.0402465
$329$	−35.1686	−1.93891
$330$	−7.00000	−0.385337
$331$	−21.1406	−1.16199	−0.580995	−	0.813907i	$-0.697337\pi$
$331$	−21.1406	−1.16199	−0.580995	+	0.813907i	$0.697337\pi$
$332$	12.1867	0.668831
$333$	−23.4859	−1.28702
$334$	0.728895	0.0398833
$335$	10.8594	0.593314
$336$	−11.4578	−0.625074
$337$	−24.5984	−1.33996	−0.669979	−	0.742380i	$-0.733697\pi$
$337$	−24.5984	−1.33996	−0.669979	+	0.742380i	$0.733697\pi$
$338$	0	0
$339$	−5.58432	−0.303299
$340$	−0.348409	−0.0188951
$341$	23.4366	1.26916
$342$	2.22188	0.120145
$343$	55.8592	3.01612
$344$	3.01404	0.162506
$345$	−13.8234	−0.744224
$346$	−15.7429	−0.846345
$347$	15.4578	0.829818	0.414909	−	0.909863i	$-0.363813\pi$
$347$	15.4578	0.829818	0.414909	+	0.909863i	$0.363813\pi$
$348$	−20.7429	−1.11194
$349$	−14.7289	−0.788420	−0.394210	−	0.919020i	$-0.628982\pi$
$349$	−14.7289	−0.788420	−0.394210	+	0.919020i	$0.628982\pi$
$350$	−17.5843	−0.939922
$351$	0	0
$352$	−2.50702	−0.133625
$353$	−23.1686	−1.23314	−0.616571	−	0.787299i	$-0.711479\pi$
$353$	−23.1686	−1.23314	−0.616571	+	0.787299i	$0.711479\pi$
$354$	5.58432	0.296803
$355$	−10.6055	−0.562880
$356$	2.50702	0.132872
$357$	−3.26710	−0.172913
$358$	−12.4890	−0.660063
$359$	18.3453	0.968228	0.484114	−	0.875005i	$-0.339142\pi$
$359$	18.3453	0.968228	0.484114	+	0.875005i	$0.339142\pi$
$360$	−2.71486	−0.143086
$361$	1.00000	0.0526316
$362$	−8.88350	−0.466907
$363$	10.7741	0.565495
$364$	0	0
$365$	14.5422	0.761174
$366$	16.0281	0.837801
$367$	23.5030	1.22685	0.613424	−	0.789754i	$-0.289792\pi$
$367$	23.5030	1.22685	0.613424	+	0.789754i	$0.289792\pi$
$368$	−4.95077	−0.258077
$369$	1.61951	0.0843085
$370$	12.9156	0.671449
$371$	1.04211	0.0541038
$372$	21.3624	1.10759
$373$	−31.4257	−1.62716	−0.813581	−	0.581452i	$-0.802485\pi$
$373$	−31.4257	−1.62716	−0.813581	+	0.581452i	$0.802485\pi$
$374$	−0.714858	−0.0369644
$375$	−23.7530	−1.22660
$376$	−7.01404	−0.361721
$377$	0	0
$378$	8.91558	0.458568
$379$	−1.23903	−0.0636446	−0.0318223	−	0.999494i	$-0.510131\pi$
$379$	−1.23903	−0.0636446	−0.0318223	+	0.999494i	$0.510131\pi$
$380$	−1.22188	−0.0626809
$381$	−9.49387	−0.486386
$382$	0.553133	0.0283008
$383$	−0.932731	−0.0476603	−0.0238302	−	0.999716i	$-0.507586\pi$
$383$	−0.932731	−0.0476603	−0.0238302	+	0.999716i	$0.507586\pi$
$384$	−2.28514	−0.116613
$385$	15.3593	0.782784
$386$	19.3273	0.983731
$387$	6.69682	0.340418
$388$	−0.478944	−0.0243147
$389$	20.7890	1.05405	0.527023	−	0.849851i	$-0.323308\pi$
$389$	20.7890	1.05405	0.527023	+	0.849851i	$0.323308\pi$
$390$	0	0
$391$	−1.41168	−0.0713915
$392$	18.1406	0.916237
$393$	−41.4859	−2.09269
$394$	2.88750	0.145470
$395$	6.28114	0.316038
$396$	−5.57028	−0.279917
$397$	25.0250	1.25597	0.627983	−	0.778227i	$-0.283881\pi$
$397$	25.0250	1.25597	0.627983	+	0.778227i	$0.283881\pi$
$398$	−14.7258	−0.738137
$399$	−11.4578	−0.573607
$400$	−3.50702	−0.175351
$401$	12.9047	0.644428	0.322214	−	0.946667i	$-0.395573\pi$
$401$	12.9047	0.644428	0.322214	+	0.946667i	$0.395573\pi$
$402$	20.3092	1.01293
$403$	0	0
$404$	−0.317220	−0.0157823
$405$	13.1094	0.651410
$406$	45.5139	2.25882
$407$	26.4999	1.31355
$408$	−0.651591	−0.0322586
$409$	−26.0561	−1.28839	−0.644197	−	0.764860i	$-0.722808\pi$
$409$	−26.0561	−1.28839	−0.644197	+	0.764860i	$0.722808\pi$
$410$	−0.890619	−0.0439845
$411$	52.6545	2.59726
$412$	5.90154	0.290748
$413$	−12.2531	−0.602934
$414$	−11.0000	−0.540621
$415$	−14.8906	−0.730952
$416$	0	0
$417$	−37.3513	−1.82910
$418$	−2.50702	−0.122622
$419$	4.98596	0.243580	0.121790	−	0.992556i	$-0.461137\pi$
$419$	4.98596	0.243580	0.121790	+	0.992556i	$0.461137\pi$
$420$	14.0000	0.683130
$421$	26.8514	1.30866	0.654329	−	0.756210i	$-0.272951\pi$
$421$	26.8514	1.30866	0.654329	+	0.756210i	$0.272951\pi$
$422$	6.60548	0.321550
$423$	−15.5843	−0.757736
$424$	0.207839	0.0100935
$425$	−1.00000	−0.0485071
$426$	−19.8343	−0.960974
$427$	−35.1686	−1.70193
$428$	−10.2078	−0.493415
$429$	0	0
$430$	−3.68278	−0.177599
$431$	12.4117	0.597849	0.298925	−	0.954277i	$-0.403372\pi$
$431$	12.4117	0.597849	0.298925	+	0.954277i	$0.403372\pi$
$432$	1.77812	0.0855500
$433$	36.7810	1.76758	0.883792	−	0.467880i	$-0.154982\pi$
$433$	36.7810	1.76758	0.883792	+	0.467880i	$0.154982\pi$
$434$	−46.8733	−2.24999
$435$	25.3453	1.21521
$436$	2.57028	0.123094
$437$	−4.95077	−0.236827
$438$	27.1967	1.29951
$439$	−10.1827	−0.485993	−0.242996	−	0.970027i	$-0.578130\pi$
$439$	−10.1827	−0.485993	−0.242996	+	0.970027i	$0.578130\pi$
$440$	3.06327	0.146035
$441$	40.3061	1.91934
$442$	0	0
$443$	11.0421	0.524627	0.262313	−	0.964983i	$-0.415515\pi$
$443$	11.0421	0.524627	0.262313	+	0.964983i	$0.415515\pi$
$444$	24.1546	1.14633
$445$	−3.06327	−0.145213
$446$	−4.41168	−0.208899
$447$	−45.4046	−2.14756
$448$	5.01404	0.236891
$449$	−13.2992	−0.627627	−0.313814	−	0.949485i	$-0.601607\pi$
$449$	−13.2992	−0.627627	−0.313814	+	0.949485i	$0.601607\pi$
$450$	−7.79216	−0.367326
$451$	−1.82735	−0.0860467
$452$	2.44375	0.114944
$453$	46.4899	2.18428
$454$	−25.8655	−1.21393
$455$	0	0
$456$	−2.28514	−0.107012
$457$	5.83739	0.273061	0.136531	−	0.990636i	$-0.456405\pi$
$457$	5.83739	0.273061	0.136531	+	0.990636i	$0.456405\pi$
$458$	−9.93673	−0.464313
$459$	0.507019	0.0236656
$460$	6.04923	0.282047
$461$	32.9085	1.53270	0.766350	−	0.642423i	$-0.222071\pi$
$461$	32.9085	1.53270	0.766350	+	0.642423i	$0.222071\pi$
$462$	28.7249	1.33640
$463$	−23.3874	−1.08690	−0.543452	−	0.839440i	$-0.682883\pi$
$463$	−23.3874	−1.08690	−0.543452	+	0.839440i	$0.682883\pi$
$464$	9.07730	0.421403
$465$	−26.1023	−1.21046
$466$	−13.5030	−0.625515
$467$	−3.30318	−0.152853	−0.0764265	−	0.997075i	$-0.524351\pi$
$467$	−3.30318	−0.152853	−0.0764265	+	0.997075i	$0.524351\pi$
$468$	0	0
$469$	−44.5623	−2.05769
$470$	8.57028	0.395318
$471$	−20.3092	−0.935799
$472$	−2.44375	−0.112483
$473$	−7.55625	−0.347437
$474$	11.7469	0.539555
$475$	−3.50702	−0.160913
$476$	1.42972	0.0655309
$477$	0.461792	0.0211440
$478$	−20.8875	−0.955372
$479$	20.9437	0.956940	0.478470	−	0.878104i	$-0.341192\pi$
$479$	20.9437	0.956940	0.478470	+	0.878104i	$0.341192\pi$
$480$	2.79216	0.127444
$481$	0	0
$482$	−23.5672	−1.07346
$483$	56.7249	2.58107
$484$	−4.71486	−0.214312
$485$	0.585210	0.0265730
$486$	19.1827	0.870144
$487$	−20.8523	−0.944908	−0.472454	−	0.881355i	$-0.656632\pi$
$487$	−20.8523	−0.944908	−0.472454	+	0.881355i	$0.656632\pi$
$488$	−7.01404	−0.317511
$489$	15.3765	0.695349
$490$	−22.1655	−1.00134
$491$	−9.68278	−0.436978	−0.218489	−	0.975839i	$-0.570113\pi$
$491$	−9.68278	−0.436978	−0.218489	+	0.975839i	$0.570113\pi$
$492$	−1.66563	−0.0750923
$493$	2.58832	0.116572
$494$	0	0
$495$	6.80620	0.305916
$496$	−9.34841	−0.419756
$497$	43.5202	1.95215
$498$	−27.8483	−1.24791
$499$	3.29918	0.147692	0.0738458	−	0.997270i	$-0.476473\pi$
$499$	3.29918	0.147692	0.0738458	+	0.997270i	$0.476473\pi$
$500$	10.3945	0.464857
$501$	−1.66563	−0.0744148
$502$	−15.8374	−0.706857
$503$	12.6344	0.563342	0.281671	−	0.959511i	$-0.409111\pi$
$503$	12.6344	0.563342	0.281671	+	0.959511i	$0.409111\pi$
$504$	11.1406	0.496240
$505$	0.387603	0.0172481
$506$	12.4117	0.551766
$507$	0	0
$508$	4.15461	0.184331
$509$	−1.17665	−0.0521541	−0.0260770	−	0.999660i	$-0.508302\pi$
$509$	−1.17665	−0.0521541	−0.0260770	+	0.999660i	$0.508302\pi$
$510$	0.796164	0.0352547
$511$	−59.6748	−2.63986
$512$	1.00000	0.0441942
$513$	1.77812	0.0785061
$514$	20.0000	0.882162
$515$	−7.21095	−0.317753
$516$	−6.88750	−0.303205
$517$	17.5843	0.773358
$518$	−52.9998	−2.32868
$519$	35.9748	1.57912
$520$	0	0
$521$	−10.7249	−0.469866	−0.234933	−	0.972012i	$-0.575487\pi$
$521$	−10.7249	−0.469866	−0.234933	+	0.972012i	$0.575487\pi$
$522$	20.1686	0.882758
$523$	−21.5952	−0.944294	−0.472147	−	0.881520i	$-0.656521\pi$
$523$	−21.5952	−0.944294	−0.472147	+	0.881520i	$0.656521\pi$
$524$	18.1546	0.793088
$525$	40.1827	1.75372
$526$	−21.1726	−0.923171
$527$	−2.66563	−0.116117
$528$	5.72889	0.249318
$529$	1.51013	0.0656580
$530$	−0.253953	−0.0110310
$531$	−5.42972	−0.235630
$532$	5.01404	0.217386
$533$	0	0
$534$	−5.72889	−0.247914
$535$	12.4727	0.539242
$536$	−8.88750	−0.383882
$537$	28.5391	1.23155
$538$	15.1406	0.652756
$539$	−45.4787	−1.95891
$540$	−2.17265	−0.0934958
$541$	−5.94677	−0.255672	−0.127836	−	0.991795i	$-0.540803\pi$
$541$	−5.94677	−0.255672	−0.127836	+	0.991795i	$0.540803\pi$
$542$	−14.1827	−0.609198
$543$	20.3001	0.871159
$544$	0.285142	0.0122254
$545$	−3.14057	−0.134527
$546$	0	0
$547$	−4.01715	−0.171761	−0.0858805	−	0.996305i	$-0.527370\pi$
$547$	−4.01715	−0.171761	−0.0858805	+	0.996305i	$0.527370\pi$
$548$	−23.0421	−0.984310
$549$	−15.5843	−0.665123
$550$	8.79216	0.374899
$551$	9.07730	0.386706
$552$	11.3132	0.481522
$553$	−25.7750	−1.09606
$554$	9.01404	0.382970
$555$	−29.5139	−1.25280
$556$	16.3453	0.693195
$557$	34.1194	1.44569	0.722843	−	0.691012i	$-0.242835\pi$
$557$	34.1194	1.44569	0.722843	+	0.691012i	$0.242835\pi$
$558$	−20.7710	−0.879307
$559$	0	0
$560$	−6.12653	−0.258893
$561$	1.63355	0.0689686
$562$	−27.5211	−1.16091
$563$	3.54913	0.149578	0.0747890	−	0.997199i	$-0.476172\pi$
$563$	3.54913	0.149578	0.0747890	+	0.997199i	$0.476172\pi$
$564$	16.0281	0.674904
$565$	−2.98596	−0.125620
$566$	−2.95789	−0.124329
$567$	−53.7951	−2.25918
$568$	8.67967	0.364191
$569$	−36.4638	−1.52864	−0.764321	−	0.644835i	$-0.776926\pi$
$569$	−36.4638	−1.52864	−0.764321	+	0.644835i	$0.776926\pi$
$570$	2.79216	0.116951
$571$	−18.8171	−0.787472	−0.393736	−	0.919224i	$-0.628818\pi$
$571$	−18.8171	−0.787472	−0.393736	+	0.919224i	$0.628818\pi$
$572$	0	0
$573$	−1.26399	−0.0528039
$574$	3.65471	0.152544
$575$	17.3624	0.724064
$576$	2.22188	0.0925782
$577$	4.63444	0.192934	0.0964671	−	0.995336i	$-0.469246\pi$
$577$	4.63444	0.192934	0.0964671	+	0.995336i	$0.469246\pi$
$578$	−16.9187	−0.703725
$579$	−44.1655	−1.83546
$580$	−11.0913	−0.460543
$581$	61.1045	2.53504
$582$	1.09446	0.0453666
$583$	−0.521056	−0.0215799
$584$	−11.9015	−0.492489
$585$	0	0
$586$	−17.6124	−0.727561
$587$	8.94365	0.369144	0.184572	−	0.982819i	$-0.440910\pi$
$587$	8.94365	0.369144	0.184572	+	0.982819i	$0.440910\pi$
$588$	−41.4538	−1.70953
$589$	−9.34841	−0.385194
$590$	2.98596	0.122930
$591$	−6.59836	−0.271420
$592$	−10.5703	−0.434436
$593$	38.3092	1.57317	0.786585	−	0.617481i	$-0.211847\pi$
$593$	38.3092	1.57317	0.786585	+	0.617481i	$0.211847\pi$
$594$	−4.45779	−0.182905
$595$	−1.74693	−0.0716173
$596$	19.8695	0.813885
$597$	33.6505	1.37722
$598$	0	0
$599$	−13.8031	−0.563979	−0.281989	−	0.959418i	$-0.590994\pi$
$599$	−13.8031	−0.563979	−0.281989	+	0.959418i	$0.590994\pi$
$600$	8.01404	0.327172
$601$	34.3373	1.40065	0.700324	−	0.713825i	$-0.253039\pi$
$601$	34.3373	1.40065	0.700324	+	0.713825i	$0.253039\pi$
$602$	15.1125	0.615939
$603$	−19.7469	−0.804157
$604$	−20.3444	−0.827802
$605$	5.76097	0.234217
$606$	0.724892	0.0294467
$607$	−27.3874	−1.11162	−0.555810	−	0.831309i	$-0.687592\pi$
$607$	−27.3874	−1.11162	−0.555810	+	0.831309i	$0.687592\pi$
$608$	1.00000	0.0405554
$609$	−104.006	−4.21453
$610$	8.57028	0.347001
$611$	0	0
$612$	0.633551	0.0256098
$613$	−34.6717	−1.40038	−0.700188	−	0.713959i	$-0.746900\pi$
$613$	−34.6717	−1.40038	−0.700188	+	0.713959i	$0.746900\pi$
$614$	−5.74693	−0.231927
$615$	2.03519	0.0820669
$616$	−12.5703	−0.506471
$617$	5.65471	0.227650	0.113825	−	0.993501i	$-0.463690\pi$
$617$	5.65471	0.227650	0.113825	+	0.993501i	$0.463690\pi$
$618$	−13.4859	−0.542481
$619$	−3.69370	−0.148462	−0.0742312	−	0.997241i	$-0.523650\pi$
$619$	−3.69370	−0.148462	−0.0742312	+	0.997241i	$0.523650\pi$
$620$	11.4226	0.458743
$621$	−8.80308	−0.353256
$622$	−15.7149	−0.630108
$623$	12.5703	0.503618
$624$	0	0
$625$	4.83427	0.193371
$626$	0.306297	0.0122421
$627$	5.72889	0.228790
$628$	8.88750	0.354650
$629$	−3.01404	−0.120178
$630$	−13.6124	−0.542331
$631$	−29.3593	−1.16878	−0.584388	−	0.811474i	$-0.698665\pi$
$631$	−29.3593	−1.16878	−0.584388	+	0.811474i	$0.698665\pi$
$632$	−5.14057	−0.204481
$633$	−15.0945	−0.599951
$634$	2.25307	0.0894807
$635$	−5.07642	−0.201451
$636$	−0.474941	−0.0188327
$637$	0	0
$638$	−22.7570	−0.900957
$639$	19.2851	0.762908
$640$	−1.22188	−0.0482989
$641$	39.2328	1.54960	0.774801	−	0.632205i	$-0.217850\pi$
$641$	39.2328	1.54960	0.774801	+	0.632205i	$0.217850\pi$
$642$	23.3264	0.920618
$643$	−12.4397	−0.490576	−0.245288	−	0.969450i	$-0.578882\pi$
$643$	−12.4397	−0.490576	−0.245288	+	0.969450i	$0.578882\pi$
$644$	−24.8234	−0.978177
$645$	8.41568	0.331367
$646$	0.285142	0.0112188
$647$	15.8844	0.624480	0.312240	−	0.950003i	$-0.398921\pi$
$647$	15.8844	0.624480	0.312240	+	0.950003i	$0.398921\pi$
$648$	−10.7289	−0.421471
$649$	6.12653	0.240487
$650$	0	0
$651$	107.112	4.19805
$652$	−6.72889	−0.263524
$653$	−43.2609	−1.69293	−0.846464	−	0.532445i	$-0.821273\pi$
$653$	−43.2609	−1.69293	−0.846464	+	0.532445i	$0.821273\pi$
$654$	−5.87347	−0.229671
$655$	−22.1827	−0.866749
$656$	0.728895	0.0284586
$657$	−26.4438	−1.03167
$658$	−35.1686	−1.37102
$659$	−1.08042	−0.0420871	−0.0210436	−	0.999779i	$-0.506699\pi$
$659$	−1.08042	−0.0420871	−0.0210436	+	0.999779i	$0.506699\pi$
$660$	−7.00000	−0.272475
$661$	3.83916	0.149326	0.0746631	−	0.997209i	$-0.476212\pi$
$661$	3.83916	0.149326	0.0746631	+	0.997209i	$0.476212\pi$
$662$	−21.1406	−0.821652
$663$	0	0
$664$	12.1867	0.472935
$665$	−6.12653	−0.237577
$666$	−23.4859	−0.910059
$667$	−44.9397	−1.74007
$668$	0.728895	0.0282018
$669$	10.0813	0.389766
$670$	10.8594	0.419536
$671$	17.5843	0.678835
$672$	−11.4578	−0.441994
$673$	5.97193	0.230201	0.115100	−	0.993354i	$-0.463281\pi$
$673$	5.97193	0.230201	0.115100	+	0.993354i	$0.463281\pi$
$674$	−24.5984	−0.947493
$675$	−6.23591	−0.240020
$676$	0	0
$677$	24.4990	0.941574	0.470787	−	0.882247i	$-0.343970\pi$
$677$	24.4990	0.941574	0.470787	+	0.882247i	$0.343970\pi$
$678$	−5.58432	−0.214465
$679$	−2.40144	−0.0921589
$680$	−0.348409	−0.0133609
$681$	59.1063	2.26496
$682$	23.4366	0.897435
$683$	31.8935	1.22037	0.610186	−	0.792258i	$-0.291095\pi$
$683$	31.8935	1.22037	0.610186	+	0.792258i	$0.291095\pi$
$684$	2.22188	0.0849556
$685$	28.1546	1.07573
$686$	55.8592	2.13272
$687$	22.7069	0.866320
$688$	3.01404	0.114909
$689$	0	0
$690$	−13.8234	−0.526246
$691$	−51.9568	−1.97653	−0.988265	−	0.152749i	$-0.951187\pi$
$691$	−51.9568	−1.97653	−0.988265	+	0.152749i	$0.951187\pi$
$692$	−15.7429	−0.598456
$693$	−27.9296	−1.06096
$694$	15.4578	0.586770
$695$	−19.9719	−0.757578
$696$	−20.7429	−0.786259
$697$	0.207839	0.00787246
$698$	−14.7289	−0.557497
$699$	30.8563	1.16709
$700$	−17.5843	−0.664625
$701$	−32.1123	−1.21286	−0.606432	−	0.795135i	$-0.707400\pi$
$701$	−32.1123	−1.21286	−0.606432	+	0.795135i	$0.707400\pi$
$702$	0	0
$703$	−10.5703	−0.398666
$704$	−2.50702	−0.0944868
$705$	−19.5843	−0.737588
$706$	−23.1686	−0.871963
$707$	−1.59055	−0.0598188
$708$	5.58432	0.209872
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	−10.6055	−0.398016
$711$	−11.4217	−0.428348
$712$	2.50702	0.0939545
$713$	46.2818	1.73327
$714$	−3.26710	−0.122268
$715$	0	0
$716$	−12.4890	−0.466735
$717$	47.7309	1.78254
$718$	18.3453	0.684640
$719$	11.1125	0.414426	0.207213	−	0.978296i	$-0.433561\pi$
$719$	11.1125	0.414426	0.207213	+	0.978296i	$0.433561\pi$
$720$	−2.71486	−0.101177
$721$	29.5906	1.10201
$722$	1.00000	0.0372161
$723$	53.8543	2.00286
$724$	−8.88350	−0.330153
$725$	−31.8343	−1.18230
$726$	10.7741	0.399865
$727$	6.57429	0.243827	0.121913	−	0.992541i	$-0.461097\pi$
$727$	6.57429	0.243827	0.121913	+	0.992541i	$0.461097\pi$
$728$	0	0
$729$	−11.6485	−0.431425
$730$	14.5422	0.538231
$731$	0.859430	0.0317872
$732$	16.0281	0.592415
$733$	2.13365	0.0788082	0.0394041	−	0.999223i	$-0.487454\pi$
$733$	2.13365	0.0788082	0.0394041	+	0.999223i	$0.487454\pi$
$734$	23.5030	0.867512
$735$	50.6514	1.86830
$736$	−4.95077	−0.182488
$737$	22.2811	0.820736
$738$	1.61951	0.0596151
$739$	14.8866	0.547613	0.273807	−	0.961785i	$-0.411717\pi$
$739$	14.8866	0.547613	0.273807	+	0.961785i	$0.411717\pi$
$740$	12.9156	0.474786
$741$	0	0
$742$	1.04211	0.0382571
$743$	−32.7701	−1.20222	−0.601110	−	0.799167i	$-0.705274\pi$
$743$	−32.7701	−1.20222	−0.601110	+	0.799167i	$0.705274\pi$
$744$	21.3624	0.783185
$745$	−24.2780	−0.889478
$746$	−31.4257	−1.15058
$747$	27.0773	0.990707
$748$	−0.714858	−0.0261378
$749$	−51.1825	−1.87017
$750$	−23.7530	−0.867336
$751$	−33.5482	−1.22419	−0.612096	−	0.790783i	$-0.709674\pi$
$751$	−33.5482	−1.22419	−0.612096	+	0.790783i	$0.709674\pi$
$752$	−7.01404	−0.255776
$753$	36.1907	1.31886
$754$	0	0
$755$	24.8583	0.904688
$756$	8.91558	0.324256
$757$	23.0060	0.836168	0.418084	−	0.908408i	$-0.362702\pi$
$757$	23.0060	0.836168	0.418084	+	0.908408i	$0.362702\pi$
$758$	−1.23903	−0.0450035
$759$	−28.3624	−1.02949
$760$	−1.22188	−0.0443221
$761$	−17.5500	−0.636188	−0.318094	−	0.948059i	$-0.603043\pi$
$761$	−17.5500	−0.636188	−0.318094	+	0.948059i	$0.603043\pi$
$762$	−9.49387	−0.343927
$763$	12.8875	0.466559
$764$	0.553133	0.0200117
$765$	−0.774121	−0.0279884
$766$	−0.932731	−0.0337009
$767$	0	0
$768$	−2.28514	−0.0824580
$769$	24.2170	0.873287	0.436643	−	0.899635i	$-0.356167\pi$
$769$	24.2170	0.873287	0.436643	+	0.899635i	$0.356167\pi$
$770$	15.3593	0.553512
$771$	−45.7028	−1.64595
$772$	19.3273	0.695603
$773$	45.5139	1.63702	0.818511	−	0.574490i	$-0.194800\pi$
$773$	45.5139	1.63702	0.818511	+	0.574490i	$0.194800\pi$
$774$	6.69682	0.240712
$775$	32.7850	1.17767
$776$	−0.478944	−0.0171931
$777$	121.112	4.34487
$778$	20.7890	0.745323
$779$	0.728895	0.0261154
$780$	0	0
$781$	−21.7601	−0.778637
$782$	−1.41168	−0.0504814
$783$	16.1406	0.576817
$784$	18.1406	0.647877
$785$	−10.8594	−0.387590
$786$	−41.4859	−1.47975
$787$	25.0782	0.893941	0.446970	−	0.894549i	$-0.352503\pi$
$787$	25.0782	0.893941	0.446970	+	0.894549i	$0.352503\pi$
$788$	2.88750	0.102863
$789$	48.3825	1.72246
$790$	6.28114	0.223473
$791$	12.2531	0.435669
$792$	−5.57028	−0.197931
$793$	0	0
$794$	25.0250	0.888103
$795$	0.580320	0.0205818
$796$	−14.7258	−0.521941
$797$	21.4569	0.760042	0.380021	−	0.924978i	$-0.375917\pi$
$797$	21.4569	0.760042	0.380021	+	0.924978i	$0.375917\pi$
$798$	−11.4578	−0.405601
$799$	−2.00000	−0.0707549
$800$	−3.50702	−0.123992
$801$	5.57028	0.196816
$802$	12.9047	0.455679
$803$	29.8374	1.05294
$804$	20.3092	0.716251
$805$	30.3311	1.06903
$806$	0	0
$807$	−34.5984	−1.21792
$808$	−0.317220	−0.0111597
$809$	42.0833	1.47957	0.739786	−	0.672843i	$-0.234927\pi$
$809$	42.0833	1.47957	0.739786	+	0.672843i	$0.234927\pi$
$810$	13.1094	0.460617
$811$	−13.0140	−0.456985	−0.228492	−	0.973546i	$-0.573380\pi$
$811$	−13.0140	−0.456985	−0.228492	+	0.973546i	$0.573380\pi$
$812$	45.5139	1.59723
$813$	32.4094	1.13665
$814$	26.4999	0.928821
$815$	8.22188	0.288000
$816$	−0.651591	−0.0228103
$817$	3.01404	0.105448
$818$	−26.0561	−0.911032
$819$	0	0
$820$	−0.890619	−0.0311018
$821$	36.7187	1.28149	0.640745	−	0.767754i	$-0.278626\pi$
$821$	36.7187	1.28149	0.640745	+	0.767754i	$0.278626\pi$
$822$	52.6545	1.83654
$823$	−9.20384	−0.320826	−0.160413	−	0.987050i	$-0.551283\pi$
$823$	−9.20384	−0.320826	−0.160413	+	0.987050i	$0.551283\pi$
$824$	5.90154	0.205590
$825$	−20.0913	−0.699491
$826$	−12.2531	−0.426339
$827$	−38.8452	−1.35078	−0.675390	−	0.737461i	$-0.736025\pi$
$827$	−38.8452	−1.35078	−0.675390	+	0.737461i	$0.736025\pi$
$828$	−11.0000	−0.382276
$829$	22.0390	0.765446	0.382723	−	0.923863i	$-0.374986\pi$
$829$	22.0390	0.765446	0.382723	+	0.923863i	$0.374986\pi$
$830$	−14.8906	−0.516861
$831$	−20.5984	−0.714549
$832$	0	0
$833$	5.17265	0.179222
$834$	−37.3513	−1.29337
$835$	−0.890619	−0.0308211
$836$	−2.50702	−0.0867071
$837$	−16.6226	−0.574562
$838$	4.98596	0.172237
$839$	39.5834	1.36657	0.683286	−	0.730151i	$-0.260550\pi$
$839$	39.5834	1.36657	0.683286	+	0.730151i	$0.260550\pi$
$840$	14.0000	0.483046
$841$	53.3974	1.84129
$842$	26.8514	0.925361
$843$	62.8895	2.16603
$844$	6.60548	0.227370
$845$	0	0
$846$	−15.5843	−0.535800
$847$	−23.6405	−0.812296
$848$	0.207839	0.00713722
$849$	6.75920	0.231975
$850$	−1.00000	−0.0342997
$851$	52.3311	1.79389
$852$	−19.8343	−0.679511
$853$	40.4959	1.38655	0.693277	−	0.720672i	$-0.256166\pi$
$853$	40.4959	1.38655	0.693277	+	0.720672i	$0.256166\pi$
$854$	−35.1686	−1.20345
$855$	−2.71486	−0.0928462
$856$	−10.2078	−0.348897
$857$	−12.5984	−0.430352	−0.215176	−	0.976575i	$-0.569032\pi$
$857$	−12.5984	−0.430352	−0.215176	+	0.976575i	$0.569032\pi$
$858$	0	0
$859$	40.5342	1.38301	0.691505	−	0.722372i	$-0.256948\pi$
$859$	40.5342	1.38301	0.691505	+	0.722372i	$0.256948\pi$
$860$	−3.68278	−0.125582
$861$	−8.35152	−0.284619
$862$	12.4117	0.422743
$863$	−37.1615	−1.26499	−0.632497	−	0.774563i	$-0.717970\pi$
$863$	−37.1615	−1.26499	−0.632497	+	0.774563i	$0.717970\pi$
$864$	1.77812	0.0604930
$865$	19.2359	0.654041
$866$	36.7810	1.24987
$867$	38.6616	1.31302
$868$	−46.8733	−1.59098
$869$	12.8875	0.437179
$870$	25.3453	0.859286
$871$	0	0
$872$	2.57028	0.0870408
$873$	−1.06415	−0.0360162
$874$	−4.95077	−0.167462
$875$	52.1185	1.76193
$876$	27.1967	0.918892
$877$	10.9780	0.370699	0.185350	−	0.982673i	$-0.440658\pi$
$877$	10.9780	0.370699	0.185350	+	0.982673i	$0.440658\pi$
$878$	−10.1827	−0.343649
$879$	40.2468	1.35749
$880$	3.06327	0.103263
$881$	22.8405	0.769516	0.384758	−	0.923017i	$-0.374285\pi$
$881$	22.8405	0.769516	0.384758	+	0.923017i	$0.374285\pi$
$882$	40.3061	1.35718
$883$	−54.7810	−1.84353	−0.921764	−	0.387750i	$-0.873252\pi$
$883$	−54.7810	−1.84353	−0.921764	+	0.387750i	$0.873252\pi$
$884$	0	0
$885$	−6.82335	−0.229364
$886$	11.0421	0.370967
$887$	29.2889	0.983427	0.491713	−	0.870757i	$-0.336371\pi$
$887$	29.2889	0.983427	0.491713	+	0.870757i	$0.336371\pi$
$888$	24.1546	0.810576
$889$	20.8314	0.698661
$890$	−3.06327	−0.102681
$891$	26.8975	0.901101
$892$	−4.41168	−0.147714
$893$	−7.01404	−0.234716
$894$	−45.4046	−1.51856
$895$	15.2600	0.510085
$896$	5.01404	0.167507
$897$	0	0
$898$	−13.2992	−0.443799
$899$	−84.8583	−2.83018
$900$	−7.79216	−0.259739
$901$	0.0592637	0.00197436
$902$	−1.82735	−0.0608442
$903$	−34.5342	−1.14923
$904$	2.44375	0.0812780
$905$	10.8545	0.360817
$906$	46.4899	1.54452
$907$	14.0452	0.466364	0.233182	−	0.972433i	$-0.425086\pi$
$907$	14.0452	0.466364	0.233182	+	0.972433i	$0.425086\pi$
$908$	−25.8655	−0.858376
$909$	−0.704823	−0.0233775
$910$	0	0
$911$	−51.2248	−1.69715	−0.848577	−	0.529073i	$-0.822540\pi$
$911$	−51.2248	−1.69715	−0.848577	+	0.529073i	$0.822540\pi$
$912$	−2.28514	−0.0756686
$913$	−30.5522	−1.01113
$914$	5.83739	0.193084
$915$	−19.5843	−0.647438
$916$	−9.93673	−0.328319
$917$	91.0279	3.00601
$918$	0.507019	0.0167341
$919$	15.2649	0.503542	0.251771	−	0.967787i	$-0.418987\pi$
$919$	15.2649	0.503542	0.251771	+	0.967787i	$0.418987\pi$
$920$	6.04923	0.199437
$921$	13.1326	0.432733
$922$	32.9085	1.08378
$923$	0	0
$924$	28.7249	0.944980
$925$	37.0702	1.21886
$926$	−23.3874	−0.768558
$927$	13.1125	0.430671
$928$	9.07730	0.297977
$929$	38.9780	1.27883	0.639413	−	0.768864i	$-0.279178\pi$
$929$	38.9780	1.27883	0.639413	+	0.768864i	$0.279178\pi$
$930$	−26.1023	−0.855927
$931$	18.1406	0.594533
$932$	−13.5030	−0.442306
$933$	35.9107	1.17566
$934$	−3.30318	−0.108083
$935$	0.873467	0.0285654
$936$	0	0
$937$	26.1718	0.854994	0.427497	−	0.904017i	$-0.359395\pi$
$937$	26.1718	0.854994	0.427497	+	0.904017i	$0.359395\pi$
$938$	−44.5623	−1.45501
$939$	−0.699932	−0.0228414
$940$	8.57028	0.279532
$941$	−54.4295	−1.77435	−0.887176	−	0.461432i	$-0.847336\pi$
$941$	−54.4295	−1.77435	−0.887176	+	0.461432i	$0.847336\pi$
$942$	−20.3092	−0.661710
$943$	−3.60859	−0.117512
$944$	−2.44375	−0.0795374
$945$	−10.8937	−0.354373
$946$	−7.55625	−0.245675
$947$	−31.7499	−1.03173	−0.515866	−	0.856669i	$-0.672530\pi$
$947$	−31.7499	−1.03173	−0.515866	+	0.856669i	$0.672530\pi$
$948$	11.7469	0.381523
$949$	0	0
$950$	−3.50702	−0.113783
$951$	−5.14858	−0.166954
$952$	1.42972	0.0463373
$953$	27.0421	0.875980	0.437990	−	0.898980i	$-0.355691\pi$
$953$	27.0421	0.875980	0.437990	+	0.898980i	$0.355691\pi$
$954$	0.461792	0.0149511
$955$	−0.675860	−0.0218703
$956$	−20.8875	−0.675550
$957$	52.0029	1.68102
$958$	20.9437	0.676659
$959$	−115.534	−3.73079
$960$	2.79216	0.0901166
$961$	56.3927	1.81912
$962$	0	0
$963$	−22.6806	−0.730871
$964$	−23.5672	−0.759047
$965$	−23.6155	−0.760210
$966$	56.7249	1.82509
$967$	−18.0561	−0.580647	−0.290323	−	0.956929i	$-0.593763\pi$
$967$	−18.0561	−0.580647	−0.290323	+	0.956929i	$0.593763\pi$
$968$	−4.71486	−0.151541
$969$	−0.651591	−0.0209321
$970$	0.585210	0.0187900
$971$	14.5732	0.467676	0.233838	−	0.972276i	$-0.424871\pi$
$971$	14.5732	0.467676	0.233838	+	0.972276i	$0.424871\pi$
$972$	19.1827	0.615285
$973$	81.9559	2.62739
$974$	−20.8523	−0.668151
$975$	0	0
$976$	−7.01404	−0.224514
$977$	15.5772	0.498359	0.249179	−	0.968457i	$-0.419839\pi$
$977$	15.5772	0.498359	0.249179	+	0.968457i	$0.419839\pi$
$978$	15.3765	0.491686
$979$	−6.28514	−0.200874
$980$	−22.1655	−0.708052
$981$	5.71085	0.182334
$982$	−9.68278	−0.308990
$983$	36.5320	1.16519	0.582595	−	0.812763i	$-0.302038\pi$
$983$	36.5320	1.16519	0.582595	+	0.812763i	$0.302038\pi$
$984$	−1.66563	−0.0530983
$985$	−3.52817	−0.112417
$986$	2.58832	0.0824291
$987$	80.3654	2.55806
$988$	0	0
$989$	−14.9218	−0.474486
$990$	6.80620	0.216315
$991$	11.9015	0.378065	0.189032	−	0.981971i	$-0.439465\pi$
$991$	11.9015	0.378065	0.189032	+	0.981971i	$0.439465\pi$
$992$	−9.34841	−0.296812
$993$	48.3092	1.53305
$994$	43.5202	1.38038
$995$	17.9931	0.570419
$996$	−27.8483	−0.882407
$997$	23.4498	0.742662	0.371331	−	0.928501i	$-0.378901\pi$
$997$	23.4498	0.742662	0.371331	+	0.928501i	$0.378901\pi$
$998$	3.29918	0.104434
$999$	−18.7953	−0.594656

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.w.1.1		3
13.12	even	2		494.2.a.f.1.1	✓	3
39.38	odd	2		4446.2.a.bk.1.2		3
52.51	odd	2		3952.2.a.o.1.3		3
247.246	odd	2		9386.2.a.bc.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.a.f.1.1	✓	3	13.12	even	2
3952.2.a.o.1.3		3	52.51	odd	2
4446.2.a.bk.1.2		3	39.38	odd	2
6422.2.a.w.1.1		3	1.1	even	1	trivial
9386.2.a.bc.1.3		3	247.246	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.28514 q^{3} +1.00000 q^{4} -1.22188 q^{5} -2.28514 q^{6} +5.01404 q^{7} +1.00000 q^{8} +2.22188 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.28514 q^{3} +1.00000 q^{4} -1.22188 q^{5} -2.28514 q^{6} +5.01404 q^{7} +1.00000 q^{8} +2.22188 q^{9} -1.22188 q^{10} -2.50702 q^{11} -2.28514 q^{12} +5.01404 q^{14} +2.79216 q^{15} +1.00000 q^{16} +0.285142 q^{17} +2.22188 q^{18} +1.00000 q^{19} -1.22188 q^{20} -11.4578 q^{21} -2.50702 q^{22} -4.95077 q^{23} -2.28514 q^{24} -3.50702 q^{25} +1.77812 q^{27} +5.01404 q^{28} +9.07730 q^{29} +2.79216 q^{30} -9.34841 q^{31} +1.00000 q^{32} +5.72889 q^{33} +0.285142 q^{34} -6.12653 q^{35} +2.22188 q^{36} -10.5703 q^{37} +1.00000 q^{38} -1.22188 q^{40} +0.728895 q^{41} -11.4578 q^{42} +3.01404 q^{43} -2.50702 q^{44} -2.71486 q^{45} -4.95077 q^{46} -7.01404 q^{47} -2.28514 q^{48} +18.1406 q^{49} -3.50702 q^{50} -0.651591 q^{51} +0.207839 q^{53} +1.77812 q^{54} +3.06327 q^{55} +5.01404 q^{56} -2.28514 q^{57} +9.07730 q^{58} -2.44375 q^{59} +2.79216 q^{60} -7.01404 q^{61} -9.34841 q^{62} +11.1406 q^{63} +1.00000 q^{64} +5.72889 q^{66} -8.88750 q^{67} +0.285142 q^{68} +11.3132 q^{69} -6.12653 q^{70} +8.67967 q^{71} +2.22188 q^{72} -11.9015 q^{73} -10.5703 q^{74} +8.01404 q^{75} +1.00000 q^{76} -12.5703 q^{77} -5.14057 q^{79} -1.22188 q^{80} -10.7289 q^{81} +0.728895 q^{82} +12.1867 q^{83} -11.4578 q^{84} -0.348409 q^{85} +3.01404 q^{86} -20.7429 q^{87} -2.50702 q^{88} +2.50702 q^{89} -2.71486 q^{90} -4.95077 q^{92} +21.3624 q^{93} -7.01404 q^{94} -1.22188 q^{95} -2.28514 q^{96} -0.478944 q^{97} +18.1406 q^{98} -5.57028 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + 4 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 - 2 * q^7 + 3 * q^8 + 4 * q^9	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + 4 q^{9} - q^{10} + q^{11} - q^{12} - 2 q^{14} - 6 q^{15} + 3 q^{16} - 5 q^{17} + 4 q^{18} + 3 q^{19} - q^{20} - 12 q^{21} + q^{22} - q^{23} - q^{24} - 2 q^{25} + 8 q^{27} - 2 q^{28} + 7 q^{29} - 6 q^{30} - 19 q^{31} + 3 q^{32} + 6 q^{33} - 5 q^{34} - 12 q^{35} + 4 q^{36} - 20 q^{37} + 3 q^{38} - q^{40} - 9 q^{41} - 12 q^{42} - 8 q^{43} + q^{44} - 14 q^{45} - q^{46} - 4 q^{47} - q^{48} + 31 q^{49} - 2 q^{50} - 11 q^{51} + 15 q^{53} + 8 q^{54} + 6 q^{55} - 2 q^{56} - q^{57} + 7 q^{58} - 2 q^{59} - 6 q^{60} - 4 q^{61} - 19 q^{62} + 10 q^{63} + 3 q^{64} + 6 q^{66} - 16 q^{67} - 5 q^{68} - 6 q^{69} - 12 q^{70} + q^{71} + 4 q^{72} - 8 q^{73} - 20 q^{74} + 7 q^{75} + 3 q^{76} - 26 q^{77} + 8 q^{79} - q^{80} - 21 q^{81} - 9 q^{82} + 3 q^{83} - 12 q^{84} + 8 q^{85} - 8 q^{86} - 34 q^{87} + q^{88} - q^{89} - 14 q^{90} - q^{92} + 38 q^{93} - 4 q^{94} - q^{95} - q^{96} - 27 q^{97} + 31 q^{98} - 5 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^5 - q^6 - 2 * q^7 + 3 * q^8 + 4 * q^9 - q^10 + q^11 - q^12 - 2 * q^14 - 6 * q^15 + 3 * q^16 - 5 * q^17 + 4 * q^18 + 3 * q^19 - q^20 - 12 * q^21 + q^22 - q^23 - q^24 - 2 * q^25 + 8 * q^27 - 2 * q^28 + 7 * q^29 - 6 * q^30 - 19 * q^31 + 3 * q^32 + 6 * q^33 - 5 * q^34 - 12 * q^35 + 4 * q^36 - 20 * q^37 + 3 * q^38 - q^40 - 9 * q^41 - 12 * q^42 - 8 * q^43 + q^44 - 14 * q^45 - q^46 - 4 * q^47 - q^48 + 31 * q^49 - 2 * q^50 - 11 * q^51 + 15 * q^53 + 8 * q^54 + 6 * q^55 - 2 * q^56 - q^57 + 7 * q^58 - 2 * q^59 - 6 * q^60 - 4 * q^61 - 19 * q^62 + 10 * q^63 + 3 * q^64 + 6 * q^66 - 16 * q^67 - 5 * q^68 - 6 * q^69 - 12 * q^70 + q^71 + 4 * q^72 - 8 * q^73 - 20 * q^74 + 7 * q^75 + 3 * q^76 - 26 * q^77 + 8 * q^79 - q^80 - 21 * q^81 - 9 * q^82 + 3 * q^83 - 12 * q^84 + 8 * q^85 - 8 * q^86 - 34 * q^87 + q^88 - q^89 - 14 * q^90 - q^92 + 38 * q^93 - 4 * q^94 - q^95 - q^96 - 27 * q^97 + 31 * q^98 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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